Slide 4b.1 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Lecture 4b Highlights of some solution methods Aspects of optimization algorithms used in topology design.

Slide 4b.2 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Contents Components of optimal synthesis Sensitivity analysis Mathematical programming algorithms An optimality criteria method Convex approximation methods

Slide 4b.3 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Optimal synthesis: what it entails Design needs  Objective function  Design variables  Constraints  Equations governing the device behavior A study to ensure the well- posedness of the optimization problem Function evaluation Sensitivity analysis Optimization algorithm Solution Satisfactory? Stop Yes No OPTIMAL SYNTHESIS

Slide 4b.4 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Sensitivity analysis Determining the gradients of the objective and constraint functions with respect to the design variables. Consider and satisfy essential (Dirichlet) boundary conditions. = smoothened state (exists or not) of a point Need to compute: (the body force is assumed to be absent.)

Slide 4b.5 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Sensitivity analysis in discrete modeling: direct method (differentiate w.r.t. ) (assuming that does not depend on ) (needs to be solved for each variable to get )

Slide 4b.6 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Sensitivity analysis in discrete modeling: adjoint method where Needs to be solved for only once! Adjoint equation (if there are constraints dependent on, then needs to be solved for those as well).

Slide 4b.7 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Sequential linear programming (SLP) Linearize Solve the LP problem and repeat until convergence. Works reasonably well, even “black-box” usage of standard packages once the problem if well formulated and understood. Especially suitable when multiple and constraints exist. Somewhat slower rates of convergence.

Slide 4b.8 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Sequential quadratic programming (SQP) “Quadratize” Solve the QP problem and repeat until convergence. Works quite well in conjunction with trust-region method (Matlab’s optimization toolbox has a routine: constr( )

Slide 4b.9 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Return to continuum model: sensitivity analysis

Slide 4b.10 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Adjoint sensitivity analysis for the continuum model Equation of equilibrium recovered from the weak form Adjoint equation (Same conclusion that we saw in slide # 2b.9 in the context of bars)

Slide 4b.11 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Optimality criteria method Adjoint. Equn. Equilib. Equn. Turned out to be the same here but not always true. Design Equn. Optimality criterion

Slide 4b.12 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Optimality criteria method: evaluating the Lagrange multiplier Check if any exceeded their upper or lower limits; if yes, limit them to the bounds. Use: Inner loop at k th iteration Repeat until do not change anymore.

Slide 4b.13 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Convex approximation methods Linearization Reciprocal linearization Convex linearization Replace with Replace with only if the partial derivative with respect to that variable is positive. Advantage: leads to convex, separable problems that can be easily solved using the more efficient dual methods (Lagrange multipliers become the variables).

Slide 4b.14 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Method of moving asymptotes (MMA) Adjustable bounds to get a conservative or accurate convex approximation of the objective and constraint expressions as necessary. K. Svanberg, “The Method of Moving Asymptotes—A New Method for Structural Optimization,” Int. J. for Num. Meth. In Engineering, Vol. 24, 1987, pp

Slide 4b.15 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Main points Function evaluation and sensitivity analysis Optimality criteria method Standard mathematical programming techniques will do (SLP, SQP) Or use convex linearization algorithms such as MMA Posing the problem correctly is crucial; most algorithms would work for properly posed problems